A consumer upset with the latest trend of postal rate increases has decided to try to send letters by balloon even though they may not reach their intended destinations. A 72800 cm3 gas-filled...

Question

A consumer upset with the latest trend of postal rate increases has decided to try to send letters by balloon even though they may not reach their intended destinations. A 72800 cm3 gas-filled balloon will provide enough lift for a 43.4 g package to be accelerated upward at a rate of 2.95 m/s2 For these circumstances, calculate the density of the gas the consumer fills the balloon with. The acceleration due to gravity is 𝑔=9.81 m/s2 and the density of air is 𝜌air=1.16 kg/m3. Neglect the mass of the balloon material and the volume of the package.

Answer 1

The buoyant force acting on the package is given by:

Buoyant force = weight of displaced air = Volume of displaced air x Density of air x g

The volume of the balloon is given as 72800 cm^3 = 0.0728 m^3. As the package is lighter than the displaced air, the buoyant force will be equal to the weight of the package. So,

Buoyant force = Weight of package = mass x g = 0.0434 kg x 2.95 m/s^2 = 0.128 N

Thus, we have:

0.128 N = 0.0728 m^3 x Density of air x g Density of gas = (0.128 N) / (0.0728 m^3 x 9.81 m/s^2 x 1.16 kg/m^3) = 0.010 kg/m^3

Therefore, the density of the gas that the consumer needs to fill the balloon with is approximately 0.01 kg/m^3. This is much lower than the density of air, which suggests that the gas must be a very light gas like helium or hydrogen.